The discussion revolves around a linear transformation from R3 to R3, specifically examining the transformation's behavior on the standard basis vectors and its implications for the range of the transformation.
Participants are actively discussing the implications of the linear transformation's definition and its representation in matrix form. Some guidance is offered regarding the nature of linear transformations and their dependence on basis vectors, but no consensus has been reached on the specific configurations of the matrices.
There is an emphasis on understanding the range of the transformation and the conditions under which different matrix representations may be valid. The discussion includes considerations of switching basis vectors and the implications for the transformation's output.
A linear transformation can be fully described by its action on any basis. Can you see why?jolly_math said:Homework Statement:: Describe explicitly a linear transformation from R3 into R3 which has as its
range the subspace spanned by (1, 0, -1) and (1, 2, 2).
Relevant Equations:: linear transformation
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?
Thank you.
Hall said:$$
\begin{bmatrix}
1 & 1 & 1 \\
0 & 0 & 2 \\
-1 & -1 & 2\\
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z\\
\end{bmatrix}
$$
Yes, I think. I think even double columns of ##(1,2,2)## will also satisfy the given things.jolly_math said:For R3, would
$$
\begin{bmatrix}
1 & 1 & 1 \\
0 & 2 & 0 \\
-1 & 2 & -1\\
\end{bmatrix}
\times
\begin{bmatrix}
x \\
y \\
z\\
\end{bmatrix}
$$
also work (switching which vector corresponds to each basis)? Thanks.